A note on error estimation for hypothesis testing problems for some linear SPDEs

The aim of the present paper is to estimate and control the Type I and Type II errors of a simple hypothesis testing problem of the drift/viscosity coefficient for stochastic fractional heat equation driven by additive noise. Assuming that one path of the first \(N\) Fourier modes of the solution is observed continuously over a finite time interval \([0,T]\), we propose a new class of rejection regions and provide computable thresholds for \(T\), and \(N\), that guarantee that the statistical errors are smaller than a given upper bound. The considered tests are of likelihood ratio type. The main ideas, and the proofs, are based on sharp large deviation bounds. Finally, we illustrate the theoretical results by numerical simulations.